Wave energy is estimated to have a potential of two terawatts worldwide. See B. Drew et al., Proc. Inst. Mech. Eng. A J. Power Energy 223(8), 887 (2009). An important aspect in harvesting wave energy is the ability to extract energy in an economic way; and hence optimizing the design and control of wave energy converters (WECs) is crucial. WEC optimization is centered on how the WEC responds to incident waves; and an important aspect of that response is the control logic. Research on WECs started in 1970s, when fundamental developments on the hydrodynamics of multi-degrees-of-freedom (MDOF) WECs were carried out. See D. V. Evans, J. Fluid Mech. 77, 1 (1976); J. N. Newman, “The interaction of stationary vessels with regular waves,” in Proceeding 11th symposium, Naval hydrodynamics, London (1976); and C. C. Mei, J. Ship Res. 20, 63 (1976). In particular, the analysis of Evans motivates the use of MDOF WECs as opposed to a single DOF WEC. For the hydrodynamics of a MDOF WEC, Vugts used velocity potential to solve for the radiation hydrodynamic coefficients and Newman applied Haskind relations to compute the excitation force. See J. H. Vugts, The hydrodynamic coefficients for swaying, heaving and rolling cylinders on a free surface, Shipbuilding Laboratory, Technical University Delft, 1968; and J. N. Newman, “The exciting forces on fixed bodies in waves,” J. Ship Res., 10 (1965). Later, several boundary element numerical tools were developed to compute the hydrodynamic coefficients for different shapes of buoys; of them are WAMIT, AQWA, OpenWARP, and NEMOH, which are frequency domain solvers that implement linear potential flow theory. See A. Babarit and G. Delhommeau, “Theoretical and numerical aspects of the open source BEM solver NEMOH,” Proceedings of the 11th European Wave and Tidal Energy Conference, Nantes, France, September 2015. In terms of the wave/body interaction, there are several references that detail the dynamics equations; of them is Korde-Ringwood and J. Falnes which give a complete development for the dynamics of six degrees-of-freedom motions resulting from buoy-wave interaction. See J. Falnes, Ocean Waves and Oscillating Systems—Linear Interactions Including Wave-Energy Extraction, Cambridge University Press, 2002.
On the controls side, however, most of the work in the literature has focused on the single DOF WEC. There have been significant developments on several concepts of control methods for single DOF WECs; many of these methods are well documented. See J. Falnes, Marine Structures 20(4), 185 (2007); U. A. Korde, OCEANS '98 Conference Proceedings 3, 1427 (1998); U. A. Korde et al., “Time domain control of a single mode wave energy device,” Proceedings of the Eleventh International Offshore and Polar Engineering Conference, Stavanger, Norway, 2001, pp. 555-560; F. Fusco and J. Ringwood, IEEE Trans. Sustain. Energy 5(3), 958 (2014); J. Ringwood et al., Control Syst. Mag. 34(5), 30 (2014); and J. Scruggs et al., Appl. Ocean Res. 42, 1 (2013). Of particular interest among these methods is the Model Predictive Control (MPC). See J. Cretel et al., IFAC Proceedings 44.1, 3714 (2011); G. Li and M. R. Belmont, Renew. Energy 69, 453 (2014); D. Oetinger et al., IEEE Trans. Sustain. Energy 5(4), 1099 (2014); M. N. Soltani et al., “Model predictive control of buoy type wave energy converter,” The 19th International Federation of Automatic Control (IFAC) World Congress, Cape Town, South Africa, Aug. 24-29, 2014; and G. Li, Int. J. Control 89, 1 (2015). In MPC, the wave is assumed to be known over a finite future horizon and the control objective is to maximize energy harvesting over this future time horizon, subject to constraints. The wave information that is available over the future horizon could be in the form of wave elevation or wave excitation force. One MPC implementation is the work by Hals et al., in which a MPC is implemented for a heave WEC device. See J. Hals et al., J. Offshore Mech. Arctic Eng. 133(1), 1 (2011). It repeatedly solves the optimization problem online in order to compute the optimal control on a receding horizon. The wave excitation force is predicted by use of an augmented Kalman filter based on a damped harmonic oscillator model of the wave process. Hals et al. investigated two different MPC objective functions. See Hals et al., J. Offshore Mech. Arctic Eng. 133(3), 031101 (2011). The first objective function is the difference between the energy entering the system and the energy radiated away from the system over the prediction horizon. The second objective function is the power absorbed by the Power Take Off unit (PTO) over the prediction horizon. Another MPC implementation is the work developed by Cretel et al. in which the objective is a refined form of maximizing the energy capture by a point absorber, while utilizing the estimation and short-term prediction of the wave excitation force. See J. Cretel et al., IFAC Proceedings 44.1, 3714 (2011). Overall, the MPC has proven to be one of the most powerful methods for energy harvesting in the single DOF point absorbers.
A MPC needs a prediction for the wave excitation force over a future time horizon. Techniques for predicting the wave elevation and using that to estimate the excitation force are well established. See A. K. Brask, “Control and estimation of wave energy converters,” MS Thesis, Norwegian University of Science and Technology, Department of Marine Technology, Norway, June 2015. For instance, Deterministic Sea Wave Prediction (DSWP) has been studied extensively in the literature where the water surface elevation is measured at one or more locations, and then those measurements are propagated forward using a wave dispersion model. Zhang et al. develop a directional hybrid wave model for short-crested irregular ocean waves for use in DSWP. See J. Zhang et al., Appl. Ocean Res. 21(4), 167 (1999). More recently, Halliday et al. investigated the use of Fast Fourier Transform (FFT) in making predictions of wave elevation and highlighted the challenges due to the periodic nature of the FFT that does not fit with the non-periodic measured signal. See J. R. Halliday et al., Renew. Energy 36(6), 1685 (2011). For wave prediction, LIDAR devices also can be used to scan the incident wave field in front of a WEC, and then propagate the measured wave field forward in space and time using a wave model for prediction. See M. R. Belmont and P. Ashwin, J. Atmospheric Ocean. Technol. 28(12), 1672 (2011). There are a number of candidate techniques that can be used for this type of problem such as the extended Kalman filter (EKF), the linear autoregressive model, and neural networks. See F. Fusco and J. V. Ringwood, IEEE Trans. Sustain. Energy 1(2), 99 (2010); and S. Hadadpour et al., “Wave energy forecasting using artificial neural networks in the Caspian sea,” Proceedings of the Institution of Civil Engineers-Maritime Engineering, vol. 167, no. 1, Institute of Civil Engineers, 2014, pp. 42-52. Recently Ling used an EKF, with a simplified linear frequency-invariant state-space model, for estimating the excitation force on a point absorber WEC, through measuring the buoy position and velocity. See B. A. Ling, “Real-time estimation and prediction of wave excitation forces for wave energy control applications,” MS Thesis, Oregon State University, June 2015; and B. A. Ling and B. A. Batten, “Real time estimation and prediction of wave excitation forces on a heaving body,” ASME 2015 34th International Conference on Ocean, Offshore and Arctic Engineering, vol. 9: Ocean Renewable Energy, no. OMAE2015-41087, St. Johns, Newfoundland, Canada: ASME, May 31-Jun. 5, 2015. The estimated excitation force is then used for predicting excitation force over short time horizons. More recently, Abdelkhalik et al. used pressure measurements to estimate the excitation force using an extended Kalman filter. See O. Abdelkhalik et al., Int. J. Control 90(8), 1793 (2016).